There are commercial tools such as:
- testng http://testng.org/doc/
- ranorex https://www.ranorex.com/web-test-automation.html
- ghostinspector https://ghostinspector.com/
- gurock testrail http://www.gurock.com/testrail/software-testing-tools.l.html
There are open source tools, primarily Selenium based tools.
- The seleniumIDE - a browser (firefox only) plugin http://www.seleniumhq.org/projects/ide/
For load testing jmeter is commonly used.
Broken links are often checked through tests which are written to check them, making sure that a 404 is not returned. There are specific tools for this such as https://go.siteimprove.com/broken-link-checker-trial-ppc-en and open source tools such as https://validator.w3.org/checklink from W3C
Spelling is normally checked manually by internal processes. There will not be regression tests for this. So many words will be trade and product specific that it takes a lot of effort to maintain automated spelling checkers for it.
Compatability of devices, browsers and versions is usually a fairly manual process. What is right for a company will depend on many company factors.